Orders of reductions of elliptic curves with many and few prime factors
2017 ◽
Vol 13
(08)
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pp. 2115-2134
Keyword(s):
In this paper, we investigate extreme values of [Formula: see text], where [Formula: see text] is an elliptic curve with complex multiplication and [Formula: see text] is the number-of-distinct-prime-divisors function. For fixed [Formula: see text], we prove an asymptotic formula for the quantity [Formula: see text]. The same result holds for the quantity [Formula: see text] when [Formula: see text]. This asymptotic formula matches what one might expect, based on a result of Delange concerning extreme values of [Formula: see text]. The argument is worked out in detail for the curve [Formula: see text], and we discuss how the method can be adapted for other CM elliptic curves.
Keyword(s):
2005 ◽
Vol 48
(1)
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pp. 16-31
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2018 ◽
Vol 2020
(24)
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pp. 10005-10041
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2005 ◽
Vol 37
(5)
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pp. 658-664
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2012 ◽
Vol 92
(1)
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pp. 45-60
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2012 ◽
Vol 149
(2)
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pp. 175-203
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2007 ◽
Vol 50
(3)
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pp. 409-417
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2015 ◽
Vol 18
(1)
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pp. 308-322
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